This paper investigates a dynamic event-triggered practically predefined-time control (PPTC) scheme for stochastic nonlinear systems (SNSs) under state constraints and input dead zones. A nonlinear state-dependent function (NSDF) is introduced to enforce state constraints. To address input dead-zone nonlinearities and mitigate the complexity growth inherent in backstepping, novel predefined-time compensating filters (PTCFs) and predefined-time filters (PTFs) are designed. These filters guarantee the convergence of filter states within a predefined time and effectively suppress the influence of dead zones. Building on this, a semiglobal predefined-time adaptive fuzzy tracking control algorithm is developed, where a fuzzy logic system (FLS) approximates unknown nonlinear dynamics. Furthermore, a dynamic event-triggered mechanism (DETM) is incorporated into the framework to reduce communication load. The present control scheme ensures tracking error convergence within a predefined time and uniform boundedness of all closed-loop signals in the pth moment. Finally, a simulation example is conducted to demonstrate the effectiveness of the proposed strategy.
{"title":"Practically predefined-time adaptive fuzzy control for stochastic nonlinear systems with full state constraints and dead zones","authors":"Mengqing Cheng , Shuo Shan , Junsheng Zhao , Shixiong Fang , Haikun Wei , Kanjian Zhang","doi":"10.1016/j.fss.2026.109779","DOIUrl":"10.1016/j.fss.2026.109779","url":null,"abstract":"<div><div>This paper investigates a dynamic event-triggered practically predefined-time control (PPTC) scheme for stochastic nonlinear systems (SNSs) under state constraints and input dead zones. A nonlinear state-dependent function (NSDF) is introduced to enforce state constraints. To address input dead-zone nonlinearities and mitigate the complexity growth inherent in backstepping, novel predefined-time compensating filters (PTCFs) and predefined-time filters (PTFs) are designed. These filters guarantee the convergence of filter states within a predefined time and effectively suppress the influence of dead zones. Building on this, a semiglobal predefined-time adaptive fuzzy tracking control algorithm is developed, where a fuzzy logic system (FLS) approximates unknown nonlinear dynamics. Furthermore, a dynamic event-triggered mechanism (DETM) is incorporated into the framework to reduce communication load. The present control scheme ensures tracking error convergence within a predefined time and uniform boundedness of all closed-loop signals in the <em>p</em>th moment. Finally, a simulation example is conducted to demonstrate the effectiveness of the proposed strategy.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109779"},"PeriodicalIF":2.7,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.fss.2026.109774
M. Svistula, T. Sribnaya, R. Uzbekov
In the present paper we propose such an abstract setting, which allows us to obtain as consequences both known and new results on the coincidence of the Choquet integral and the pan-integral. For example: in the case of a measurable space we derive a well-known theorem that the weak (M)-property of a monotone measure is necessary and sufficient for the coincidence of the integrals under consideration for all nonnegative measurable integrands; in the case of a topological space we use the integrals with respect to a regular monotone measure and establish some new results, in particular, that the Choquet integral and the pan-integral with respect to a topological measure coincide for all nonnegative lower semicontinuous integrands.
Next, in the case of a measurable space we give an example to show that the weak (M)-property is weaker than the middle (M)-property, and thus we solve an open problem of the relationship between these properties.
{"title":"On the coincidence of the Choquet integral and the pan-integral: An abstract setting and examples","authors":"M. Svistula, T. Sribnaya, R. Uzbekov","doi":"10.1016/j.fss.2026.109774","DOIUrl":"10.1016/j.fss.2026.109774","url":null,"abstract":"<div><div>In the present paper we propose such an abstract setting, which allows us to obtain as consequences both known and new results on the coincidence of the Choquet integral and the pan-integral. For example: in the case of a measurable space we derive a well-known theorem that the weak (M)-property of a monotone measure is necessary and sufficient for the coincidence of the integrals under consideration for all nonnegative measurable integrands; in the case of a topological space we use the integrals with respect to a regular monotone measure and establish some new results, in particular, that the Choquet integral and the pan-integral with respect to a topological measure coincide for all nonnegative lower semicontinuous integrands.</div><div>Next, in the case of a measurable space we give an example to show that the weak (M)-property is weaker than the middle (M)-property, and thus we solve an open problem of the relationship between these properties.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"532 ","pages":"Article 109774"},"PeriodicalIF":2.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146049214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.fss.2026.109772
Chuang Zheng
<div><div>In this paper, we solve the fuzzy linear systems in a fuzzy number space <span><math><mi>X</mi></math></span>, namely the Gaussian probability density membership function (Gaussian-PDMF) space. The fuzzy linear systems include two types: the semi-fuzzy linear system (SFLS) and the fully-fuzzy linear system (FFLS). First, we solve the SFLS <span><math><mrow><mi>A</mi><mrow><mover><mi>x</mi><mo>˜</mo></mover></mrow><mo>=</mo><mrow><mover><mi>b</mi><mo>˜</mo></mover></mrow></mrow></math></span>, where <span><math><mrow><mi>A</mi><mo>∈</mo><msup><mi>R</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></math></span> is a real-valued matrix, <span><math><mrow><mover><mi>b</mi><mo>˜</mo></mover></mrow></math></span> is a fuzzy number vector, and <span><math><mrow><mover><mi>x</mi><mo>˜</mo></mover></mrow></math></span> is the unknown fuzzy number vector. The elements of both <span><math><mrow><mover><mi>b</mi><mo>˜</mo></mover></mrow></math></span> and <span><math><mrow><mover><mi>x</mi><mo>˜</mo></mover></mrow></math></span> belong to <span><math><mi>X</mi></math></span>. We present the Cramer’s rule to calculate the solution with square matrix <em>A</em> and find out that its solution set is a <span><math><mrow><mn>5</mn><mo>(</mo><mi>n</mi><mo>−</mo><mi>R</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>)</mo></mrow></math></span> dimensional affine space with <span><math><mrow><mi>A</mi><mo>∈</mo><msup><mi>R</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></math></span> and <em>R</em>(<em>A</em>) being the rank of <em>A</em>. The explicit form of the solution for RREF matrix <em>A</em> is stated to ensure usability for modeling. Secondly, we solve the FFLS <span><math><mrow><mrow><mover><mi>A</mi><mo>˜</mo></mover></mrow><mrow><mover><mi>x</mi><mo>˜</mo></mover></mrow><mo>=</mo><mrow><mover><mi>b</mi><mo>˜</mo></mover></mrow></mrow></math></span>, where <span><math><mrow><mover><mi>A</mi><mo>˜</mo></mover></mrow></math></span> is a fuzzy matrix with all components in <span><math><mi>X</mi></math></span>. We analyze its solution set and present the parametric form of solutions under the fuzzy RREF matrix. We then adapt Gaussian elimination method to fuzzy matrices and systems by restricting it to the unit group of ring <span><math><mi>X</mi></math></span>, proving the equivalence of solution sets after elementary row operations. We also establish the connection between FFLS and SFLS by confining elements of <span><math><mrow><mover><mi>A</mi><mo>˜</mo></mover></mrow></math></span> to a subset of <span><math><mi>X</mi></math></span> that forms a field. In the third part, two numerical examples are given to illustrated our method. All results in this paper are explicit since the Gaussian-PDMF space <span><math><mi>X</mi></math></span>, to which the membership function of the fuzzy number belongs, possesses a complete algebraic structure. The proposed framework offers a feasible and systematical tool for solving the mathematical m
本文在模糊数空间X，即高斯概率密度隶属函数（Gaussian- pdmf）空间中求解模糊线性系统。模糊线性系统包括半模糊线性系统和全模糊线性系统两种类型。首先，我们求解SFLS Ax ~ =b ~，其中A∈Rm×n为实值矩阵，b ~为模糊数向量，x ~为未知模糊数向量。我们提出了计算具有方阵A的解的Cramer规则，并发现其解集是一个5（n−R(A)）维仿射空间，其中A∈Rm×n， R(A)为A的秩。为了保证建模的可用性，我们给出了RREF矩阵A解的显式形式。其次，我们求解了FFLS A ~ x ~ =b ~，其中A ~是一个所有成分都在x中的模糊矩阵，我们分析了它的解集，并给出了模糊RREF矩阵下解的参数形式。然后将高斯消去法限定在环X的单位群上，将其应用于模糊矩阵和系统，证明了初等行运算后解集的等价性。我们还通过将A ~的元素限定为X的一个子集来建立FFLS和SFLS之间的联系。在第三部分中，给出了两个数值例子来说明我们的方法。由于模糊数的隶属函数所在的高斯- pdmf空间X具有完备的代数结构，所以本文的所有结果都是显式的。该框架为求解具有不确定性和模糊性的模糊线性系统的数学模型提供了一种可行的系统工具。
{"title":"Solving fuzzy linear systems in Gaussian PDMF space","authors":"Chuang Zheng","doi":"10.1016/j.fss.2026.109772","DOIUrl":"10.1016/j.fss.2026.109772","url":null,"abstract":"<div><div>In this paper, we solve the fuzzy linear systems in a fuzzy number space <span><math><mi>X</mi></math></span>, namely the Gaussian probability density membership function (Gaussian-PDMF) space. The fuzzy linear systems include two types: the semi-fuzzy linear system (SFLS) and the fully-fuzzy linear system (FFLS). First, we solve the SFLS <span><math><mrow><mi>A</mi><mrow><mover><mi>x</mi><mo>˜</mo></mover></mrow><mo>=</mo><mrow><mover><mi>b</mi><mo>˜</mo></mover></mrow></mrow></math></span>, where <span><math><mrow><mi>A</mi><mo>∈</mo><msup><mi>R</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></math></span> is a real-valued matrix, <span><math><mrow><mover><mi>b</mi><mo>˜</mo></mover></mrow></math></span> is a fuzzy number vector, and <span><math><mrow><mover><mi>x</mi><mo>˜</mo></mover></mrow></math></span> is the unknown fuzzy number vector. The elements of both <span><math><mrow><mover><mi>b</mi><mo>˜</mo></mover></mrow></math></span> and <span><math><mrow><mover><mi>x</mi><mo>˜</mo></mover></mrow></math></span> belong to <span><math><mi>X</mi></math></span>. We present the Cramer’s rule to calculate the solution with square matrix <em>A</em> and find out that its solution set is a <span><math><mrow><mn>5</mn><mo>(</mo><mi>n</mi><mo>−</mo><mi>R</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>)</mo></mrow></math></span> dimensional affine space with <span><math><mrow><mi>A</mi><mo>∈</mo><msup><mi>R</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup></mrow></math></span> and <em>R</em>(<em>A</em>) being the rank of <em>A</em>. The explicit form of the solution for RREF matrix <em>A</em> is stated to ensure usability for modeling. Secondly, we solve the FFLS <span><math><mrow><mrow><mover><mi>A</mi><mo>˜</mo></mover></mrow><mrow><mover><mi>x</mi><mo>˜</mo></mover></mrow><mo>=</mo><mrow><mover><mi>b</mi><mo>˜</mo></mover></mrow></mrow></math></span>, where <span><math><mrow><mover><mi>A</mi><mo>˜</mo></mover></mrow></math></span> is a fuzzy matrix with all components in <span><math><mi>X</mi></math></span>. We analyze its solution set and present the parametric form of solutions under the fuzzy RREF matrix. We then adapt Gaussian elimination method to fuzzy matrices and systems by restricting it to the unit group of ring <span><math><mi>X</mi></math></span>, proving the equivalence of solution sets after elementary row operations. We also establish the connection between FFLS and SFLS by confining elements of <span><math><mrow><mover><mi>A</mi><mo>˜</mo></mover></mrow></math></span> to a subset of <span><math><mi>X</mi></math></span> that forms a field. In the third part, two numerical examples are given to illustrated our method. All results in this paper are explicit since the Gaussian-PDMF space <span><math><mi>X</mi></math></span>, to which the membership function of the fuzzy number belongs, possesses a complete algebraic structure. The proposed framework offers a feasible and systematical tool for solving the mathematical m","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109772"},"PeriodicalIF":2.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.fss.2026.109773
Hao Qiu , Huamin Wang , Likui Wang , Shiping Wen
In practical industrial systems, we often encounter nonlinear uncertainties, parameter jumps, or time-delay phenomena that easily destroy the stability of the system. To stabilize these disturbances, we construct a more general closed-loop interval type-2 fuzzy delayed semi-Markov jump system (IT-2FD S-MJS) and investigate its stochastic stability in this article. Firstly, to optimize the performance of computational resources and data transmission, we introduce quantization techniques and novel adaptive dynamic data sampling-based event-triggered mechanisms, greatly reducing the number of triggers and improving the flexibility of adjustment. The framework’s discrete sampling nature inherently prevents Zeno behavior by eliminating the possibility of infinite triggers within finite time intervals. Then, by constructing boundary/general uncertain transition rates (BUTR/GUTR) and slack matrices, we derive sufficient conditions with less conservatism of stochastic stability for IT-2FD S-MJS. It should be noticed that the unknown transition information is modeled by BUTR/GUTR, and the conservatism of mismatched membership functions is reduced by introducing the slack matrices. Meanwhile, we obtain the corresponding gain parameters of the adaptive dynamic event-triggered quantization controller using linear matrix inequality (LMI) technology, with implementation details specified in Algorithms 1 and 2. Finally, we take the robotic arm and tunnel diode circuit as examples to verify the validity of the theorems.
{"title":"Adaptive dynamic event-triggered control for IT-2 fuzzy delayed semi-Markov jump systems with different uncertain transition rates","authors":"Hao Qiu , Huamin Wang , Likui Wang , Shiping Wen","doi":"10.1016/j.fss.2026.109773","DOIUrl":"10.1016/j.fss.2026.109773","url":null,"abstract":"<div><div>In practical industrial systems, we often encounter nonlinear uncertainties, parameter jumps, or time-delay phenomena that easily destroy the stability of the system. To stabilize these disturbances, we construct a more general closed-loop interval type-2 fuzzy delayed semi-Markov jump system (IT-2FD S-MJS) and investigate its stochastic stability in this article. Firstly, to optimize the performance of computational resources and data transmission, we introduce quantization techniques and novel adaptive dynamic data sampling-based event-triggered mechanisms, greatly reducing the number of triggers and improving the flexibility of adjustment. The framework’s discrete sampling nature inherently prevents Zeno behavior by eliminating the possibility of infinite triggers within finite time intervals. Then, by constructing boundary/general uncertain transition rates (BUTR/GUTR) and slack matrices, we derive sufficient conditions with less conservatism of stochastic stability for IT-2FD S-MJS. It should be noticed that the unknown transition information is modeled by BUTR/GUTR, and the conservatism of mismatched membership functions is reduced by introducing the slack matrices. Meanwhile, we obtain the corresponding gain parameters of the adaptive dynamic event-triggered quantization controller using linear matrix inequality (LMI) technology, with implementation details specified in Algorithms 1 and 2. Finally, we take the robotic arm and tunnel diode circuit as examples to verify the validity of the theorems.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109773"},"PeriodicalIF":2.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-13DOI: 10.1016/j.fss.2026.109768
Yang Liu , Zhen Wang , Xia Huang , Hao Shen
This paper put forward a type of general discontinuous activation functions (AFs) and then investigate the ψ-type multistability of fuzzy neural networks (FNNs). Through determining some algebraic inequalities, it is shown that FNNs with such discontinuous AFs can produce equilibrium points (EPs), in which EPs are locally ψ-stable and located at points of continuity (POC) of the AFs. Here, k refers to the number of discontinuous points of the AFs. Depending on the choice of the function ψ(t), the obtained EPs in FNNs can exhibit different types of stability. FNNs with the designed AFs are able to possess larger number of locally ψ-stable EPs and total EPs compared with general continuous AFs. Therefore, when applied in associative memory, FNNs with the above discontinuous AFs are able to store more memory patterns. Besides, attraction basins (ABs) associated with the ψ-stable EPs in FNNs are estimated. The correctness of the obtained results are verified through three examples.
{"title":"ψ-Type multistability of takagi-Sugeno fuzzy neural networks with general discontinuous activation functions","authors":"Yang Liu , Zhen Wang , Xia Huang , Hao Shen","doi":"10.1016/j.fss.2026.109768","DOIUrl":"10.1016/j.fss.2026.109768","url":null,"abstract":"<div><div>This paper put forward a type of general discontinuous activation functions (AFs) and then investigate the <em>ψ</em>-type multistability of fuzzy neural networks (FNNs). Through determining some algebraic inequalities, it is shown that FNNs with such discontinuous AFs can produce <span><math><msup><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup></math></span> equilibrium points (EPs), in which <span><math><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup></math></span> EPs are locally <em>ψ</em>-stable and located at points of continuity (POC) of the AFs. Here, <em>k</em> refers to the number of discontinuous points of the AFs. Depending on the choice of the function <em>ψ</em>(<em>t</em>), the obtained <span><math><msup><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup></math></span> EPs in FNNs can exhibit different types of stability. FNNs with the designed AFs are able to possess larger number of locally <em>ψ</em>-stable EPs and total EPs compared with general continuous AFs. Therefore, when applied in associative memory, FNNs with the above discontinuous AFs are able to store more memory patterns. Besides, attraction basins (ABs) associated with the <em>ψ</em>-stable EPs in FNNs are estimated. The correctness of the obtained results are verified through three examples.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109768"},"PeriodicalIF":2.7,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-13DOI: 10.1016/j.fss.2026.109775
James C. Bezdek, Thomas A. Runkler
Possibilistic c-means (PCM) clustering began in 1993, and has been used since then in many applications. In this article we discuss the geometric foundations of PCM and introduce a new hard possibilistic c-means (HPCM) clustering algorithm. We use limit theory to prove that the extended set of possibilistic c-partitions is the unit hypercube in; and that its vertices are exactly the hard possibilistic c-partitions on n objects defined herein. This enables completion of the geometric description of the domain of possibilistic clustering algorithms. We give examples that compare the results of clustering with Hard c-means (HCM) to HPCM on three small synthetic data sets. Our proof-of-concept examples show that the new algorithm performs as expected, and provides much more realistic interpretation of clusters than HCM when the data contain bridge points or noise.
{"title":"Geometric foundations of possibilistic clustering: A hard possibilistic clustering algorithm","authors":"James C. Bezdek, Thomas A. Runkler","doi":"10.1016/j.fss.2026.109775","DOIUrl":"10.1016/j.fss.2026.109775","url":null,"abstract":"<div><div><em>Possibilistic c-means</em> (PCM) clustering began in 1993, and has been used since then in many applications. In this article we discuss the geometric foundations of PCM and introduce a new <em>hard possibilistic c-means</em> (HPCM) clustering algorithm. We use limit theory to prove that the extended set of possibilistic c-partitions is the unit hypercube in<span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>c</mi><mi>n</mi></mrow></msup></math></span>; and that its vertices are exactly the hard possibilistic c-partitions on n objects defined herein. This enables completion of the geometric description of the domain of possibilistic clustering algorithms. We give examples that compare the results of clustering with <em>Hard c-means</em> (HCM) to HPCM on three small synthetic data sets. Our proof-of-concept examples show that the new algorithm performs as expected, and provides much more realistic interpretation of clusters than HCM when the data contain bridge points or noise.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"532 ","pages":"Article 109775"},"PeriodicalIF":2.7,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146049216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.fss.2026.109770
Kejin Li, Feifei Du
The projective synchronization (PS) of discrete-time fractional-order fuzzy cellular neural networks (DFFCNNs) with distributed delays is investigated in this paper. First, based on the nabla fractional-order difference theory, a comparison principle suitable for fractional-order systems with variable coefficients and multiple time delays is established, and the sub-multiplicative law of the nabla Mittag-Leffler function is rigorously proved. Second, a discrete-time fractional-order Halanay inequality with arbitrary step size, variable coefficients, and multiple time-varying delays is introduced. Furthermore, leveraging the aforementioned inequality, a sufficient condition for the PS of DFFCNNs is derived. Finally, an example is presented to confirm the validity of the results.
{"title":"Exploring projective synchronization in discrete-time fractional-order fuzzy cellular neural networks with distributed delays","authors":"Kejin Li, Feifei Du","doi":"10.1016/j.fss.2026.109770","DOIUrl":"10.1016/j.fss.2026.109770","url":null,"abstract":"<div><div>The projective synchronization (PS) of discrete-time fractional-order fuzzy cellular neural networks (DFFCNNs) with distributed delays is investigated in this paper. First, based on the nabla fractional-order difference theory, a comparison principle suitable for fractional-order systems with variable coefficients and multiple time delays is established, and the sub-multiplicative law of the nabla Mittag-Leffler function is rigorously proved. Second, a discrete-time fractional-order Halanay inequality with arbitrary step size, variable coefficients, and multiple time-varying delays is introduced. Furthermore, leveraging the aforementioned inequality, a sufficient condition for the PS of DFFCNNs is derived. Finally, an example is presented to confirm the validity of the results.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109770"},"PeriodicalIF":2.7,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145963130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.fss.2026.109769
Siyu Xu, Xiaodong Pan, Yexing Dan, Keyun Qin
Recently, by using overlap and grouping functions, Han et al. introduced two novel types of fuzzy rough sets on complete lattices in an L-fuzzy approximation space (U, V, R), along with their application to three-way decisions. We refer to these two types of fuzzy rough sets as the 1st type and the 2nd type of fuzzy rough sets. It is worth noting that, in the case where (U, V, R) is an L-fuzzy approximation space, many properties of these two types of fuzzy rough sets established in L-fuzzy approximation spaces of the form (U, R) (i.e., ) are generally difficult to establish in this more general framework. Therefore, in this paper, we focus on exploring some properties of these two types of fuzzy rough sets under the restriction to an L-fuzzy approximation space (U, R), with particular emphasis on how they generate Alexandrov L-fuzzy topologies. In particular, regarding the 2nd type of fuzzy rough sets, we deduce the behaviours of the upper and lower L-fuzzy rough approximation operators of an L-fuzzy approximation space (U, R) in the case of a family of L-fuzzy relations. Moreover, we explore the relationships among the pair of upper and lower L-fuzzy rough approximation operators proposed by Jiang and Hu in 2022 and the two pairs of upper and lower L-fuzzy rough approximation operators introduced in this study. Our investigations can be regarded as a contribution to enriching the theoretical framework of the two novel types of fuzzy rough sets by Han et al. in an L-fuzzy approximation space (U, V, R).
{"title":"Some properties of two types of fuzzy rough sets on complete lattices constructed by means of overlap and grouping functions","authors":"Siyu Xu, Xiaodong Pan, Yexing Dan, Keyun Qin","doi":"10.1016/j.fss.2026.109769","DOIUrl":"10.1016/j.fss.2026.109769","url":null,"abstract":"<div><div>Recently, by using overlap and grouping functions, Han et al. introduced two novel types of fuzzy rough sets on complete lattices in an <em>L</em>-fuzzy approximation space (<em>U, V, R</em>), along with their application to three-way decisions. We refer to these two types of fuzzy rough sets as the 1<sup><em>st</em></sup> type and the 2<sup><em>nd</em></sup> type of fuzzy rough sets. It is worth noting that, in the case where (<em>U, V, R</em>) is an <em>L</em>-fuzzy approximation space, many properties of these two types of fuzzy rough sets established in <em>L</em>-fuzzy approximation spaces of the form (<em>U, R</em>) (i.e., <span><math><mrow><mi>U</mi><mo>=</mo><mi>V</mi></mrow></math></span>) are generally difficult to establish in this more general framework. Therefore, in this paper, we focus on exploring some properties of these two types of fuzzy rough sets under the restriction to an <em>L</em>-fuzzy approximation space (<em>U, R</em>), with particular emphasis on how they generate Alexandrov <em>L</em>-fuzzy topologies. In particular, regarding the 2<sup><em>nd</em></sup> type of fuzzy rough sets, we deduce the behaviours of the upper and lower <em>L</em>-fuzzy rough approximation operators of an <em>L</em>-fuzzy approximation space (<em>U, R</em>) in the case of a family of <em>L</em>-fuzzy relations. Moreover, we explore the relationships among the pair of upper and lower <em>L</em>-fuzzy rough approximation operators proposed by Jiang and Hu in 2022 and the two pairs of upper and lower <em>L</em>-fuzzy rough approximation operators introduced in this study. Our investigations can be regarded as a contribution to enriching the theoretical framework of the two novel types of fuzzy rough sets by Han et al. in an <em>L</em>-fuzzy approximation space (<em>U, V, R</em>).</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109769"},"PeriodicalIF":2.7,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.fss.2026.109767
Lili Shen, Jian Zhang
Let [0, 1]* be the unit interval [0,1] equipped with a continuous t-norm *. It is shown that the category of [0, 1]*-sets is cartesian closed if, and only if, * is the minimum t-norm on [0,1].
{"title":"Cartesian closedness of the category of real-valued sets, I","authors":"Lili Shen, Jian Zhang","doi":"10.1016/j.fss.2026.109767","DOIUrl":"10.1016/j.fss.2026.109767","url":null,"abstract":"<div><div>Let [0, 1]<sub>*</sub> be the unit interval [0,1] equipped with a continuous t-norm *. It is shown that the category of [0, 1]<sub>*</sub>-sets is cartesian closed if, and only if, * is the minimum t-norm on [0,1].</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109767"},"PeriodicalIF":2.7,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-10DOI: 10.1016/j.fss.2026.109771
Roberto G. Aragón, Jesús Medina, Samuel Molina-Ruiz
In many situations is fundamental to use a procedure to aggregate information obtained from different sources (devices), such as when an edge computing system is used. Bonds were introduced in formal concept analysis as an aggregation method for linking different contexts (datasets) whilst preserving the information they contain. In this paper, we generalize the notion of bond to the multi-adjoint concept lattice framework, which is a fuzzy and flexible extension of formal concept analysis. Furthermore, we study several properties of multi-adjoint bonds defined by the constantly top or constantly bottom relations, with an emphasis on how they aggregate the information in the concept lattices.
{"title":"Context-based sum via multi-adjoint bonds","authors":"Roberto G. Aragón, Jesús Medina, Samuel Molina-Ruiz","doi":"10.1016/j.fss.2026.109771","DOIUrl":"10.1016/j.fss.2026.109771","url":null,"abstract":"<div><div>In many situations is fundamental to use a procedure to aggregate information obtained from different sources (devices), such as when an edge computing system is used. Bonds were introduced in formal concept analysis as an aggregation method for linking different contexts (datasets) whilst preserving the information they contain. In this paper, we generalize the notion of bond to the multi-adjoint concept lattice framework, which is a fuzzy and flexible extension of formal concept analysis. Furthermore, we study several properties of multi-adjoint bonds defined by the constantly top or constantly bottom relations, with an emphasis on how they aggregate the information in the concept lattices.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"531 ","pages":"Article 109771"},"PeriodicalIF":2.7,"publicationDate":"2026-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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